#626910

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,19)(20,24,21,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38);(2,5)(6,8)(9,11,14,13)(10,15,12,16)(17,19,22,18)(20,24,21,23)(26,27)(28,35)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,13,16)(17,20,22,21)(18,23,19,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,21,22,20)(18,24,19,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 534106);